A number of methods have heretofore been proposed for the production of carbon fibers from acrylonitrile-based fibers. These methods have been developed for various purposes, for example, improvements in the chemical and physical properties of carbon fibers and rationalization (improving efficiency) of production steps. With regard to the improvements in the physical properties, many of the proposed methods are directed to improvements in the tensile strength and tensile modulus of carbon fibers. In many cases, carbon fibers are in practical use as composite materials in combination with resins such as an epoxy resin. Composite materials prepared using conventional carbon fibers, however, are inferior in impact strength, although they are excellent in the tensile and flexural strengths.